8 CLOSED SETS, CLUSTER POINTS, CLOSURE OF A SET 39

Suppose the contrary, and let V n A = {y1}.. .,yn}: by assumption,
rk = d(x, yk) > 0, Let r > 0 be such that E(x; r) cz V and r < min^,..., rk);
then the intersection of A and E(x; r) would be empty, contrary to assumption.

A point x 6 E is said to be a frontier point of a set A if it is a cluster point
of both A and (J A; the set fr(A) of all frontier points of A is called the
frontier of A. It is clear that fr(A) = A n ^A = fr((J A); by (3.8.6), fr(A) is a
closed set, which may be empty (see (3.19.9)). A frontier point x of A is
characterized by the property that in any neighborhood of x there is at least
one point of A and one point of (J A. The whole space E is the union of the
interior of A, the exterior of A and the frontier of A, for if a neighborhood
of x is neither contained in A nor in (j A, it must contain points of both; any
two of these three sets have no common points.

The frontier of any interval of origin a and extremity b in R is the set
{a, b}; the frontier of the set Q in R is R itself.

PROBLEMS

1. (a) Let A be an open set in a metric space E; show that for any subset B of E,
A n B c A n B.

(b) Give examples in the real line, of open sets A, B such that the four sets A n B,
B n A, A n B and A n B are all different.

(c) Give an example of two intervals A, B in the real line, such that A n B is not
contained in A n B.

o —

2. For every subset A of a metric space E, let oc(A) = A and /?(A) = A.

(a) Show that if A is open, A c a(A), and if A is closed, A => /?(A).

(b) Show that for every subset A of E, a(a(A)) = a(A) and £(0(A)) = j3(A) (use (a)).

(c) Give an example, in the real line, of a set A such that the seven sets A, A, A,
a(A), /3(A), a(A), )8(A) are all distinct and have no other inclusion relations than the
following ones: A c A c A; A c a(A) <= 0(A) c A, A c a(A) c /?(A) c A.

3. Let E be a metric space.

(a) Show that for every subset A of E, fr(A) c fr(A), fr(A) <= fr(A), and give examples
(in the real line) in which these three sets are distinct.

(b) Let A, B be two subsets of E. Show that fr(A u B) <= fr(A) u fr(B), and give an
example (in the real line) in which these sets are distinct. If A n B « 0, show that
fr(A u B) = fr(A) u fr(B).

(c) If A and B are open, show that

(A n fr(B)) u (B n fr(A)) c fr(A n B) c (A n fr(B)) vj (B n fr(A)) u (fr(A) n fr(B))

and give an example (in the real line) in which these three sets are distinct.

4. Let d be a distance on a set E, satisfying the ultrametric inequality

d(x, z) ^ max(d(x, y), d(yt z))
for AT, y, z in E (see Example (3.2.6)).et of elements.it possible for me to teach this course along the lines I had planned